nLab
Reeb graph
Contents
Context
Topology
topology (point-set topology , point-free topology )

see also differential topology , algebraic topology , functional analysis and topological homotopy theory

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents
Definition
Given a topological space $X$ equipped with a continuous function $f \,\colon\, X \longrightarrow \mathbb{R}^1$ to the Euclidean space of real numbers , it’s Reeb graph is the quotient topological space of $X$ by the equivalence relation which regards two points $x, y \,\in\, X$ as equivalent iff they are in the same connected component of the same level set . Under good conditions this is a CW complex , necessarily 1-dimensional , and as such an undirected graph .

References
Original article:

Georges Reeb , Sur les points singuliers d’une forme de Pfaff completement integrable ou d’une fonction numerique , Comptes Rendus Acad. Sciences Paris 222 (1946) 847-849 $[$ crid:1571417125676878592 $]$
Further developments:

See also:

Last revised on May 26, 2022 at 15:54:40.
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